Functional principal component analysis (FPCA) is a statistical method for investigating the dominant modes of variation of functional data. Using this method, a random function is represented in the eigenbasis, which is an orthonormal basis of the Hilbert space L2 that consists of the eigenfunctions of the autocovariance operator. FPCA represents functional data in the most parsimonious way, in the sense that when using a fixed number of basis functions, the eigenfunction basis explains more variation than any other basis expansion. FPCA can be applied for representing random functions, or in functional regression and classification.
Formulation For a square-integrable stochastic process X(t), t ∈ 𝒯, let
μ ( t ) = E ( X ( t ) ) {\displaystyle \mu (t)={\text{E}}(X(t))}
and
G ( s , t ) = Cov ( X ( s ) , X ( t ) ) = ∑ k = 1 ∞ λ k φ k ( s ) φ k ( t ) , {\displaystyle G(s,t)={\text{Cov}}(X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),}
where λ 1 ≥ λ 2 ≥ . . . ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq ...\geq 0} are the eigenvalues and φ 1 {\displaystyle \varphi _{1}} , φ 2 {\displaystyle \varphi _{2}} , ... are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator
G : L 2 ( T ) → L 2 ( T ) , G ( f ) = ∫ T G ( s , t ) f ( s ) d s . {\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.}
By the Karhunen–Loève theorem, one can express the centered process in the eigenbasis,
X ( t ) − μ ( t ) = ∑ k = 1 ∞ ξ k φ k ( t ) , {\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),}
where
ξ k = ∫ T ( X ( t ) − μ ( t ) ) φ k ( t ) d t {\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt}
is the principal component associated with the k-th eigenfunction φ k {\displaystyle \varphi _{k}} , with the properties
E ( ξ k ) = 0 , Var ( ξ k ) = λ k and E ( ξ k ξ l ) = 0 for k ≠ l . {\displaystyle {\text{E}}(\xi _{k})=0,{\text{Var}}(\xi _{k})=\lambda _{k}{\text{ and }}{\text{E}}(\xi _{k}\xi _{l})=0{\text{ for }}k\neq l.}
The centered process is then equivalent to ξ1, ξ2, .... A common assumption is that X can be represented by only the first few eigenfunctions (after subtracting the mean function), i.e.
X ( t ) ≈ X m ( t ) = μ ( t ) + ∑ k = 1 m ξ k φ k ( t ) , {\displaystyle X(t)\approx X_{m}(t)=\mu (t)+\sum _{k=1}^{m}\xi _{k}\varphi _{k}(t),}
where
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